I have saved the text from an article by Pat Puckett that was published by Western Horseman in 2003; the article describes the care of a reata pretty well. Western Horseman's website doesn't have the article available online anymore, but I´d be happy to email a copy of what I have to you if you can live without the pictures of the original article. Contact me pm with your email and I´ll be happy to get the article to you (or anyone else interested).
If the reata is new, you won't need to stretch it so much as you will need to "pull" it to not only get the reata stretched, but to get the reata greased all the way to the core, as well. This will soften a stiff, new reata, and get it "usable". But if your reata has been used, just a good cleaning and greasing should work.
Long story short, and I´m quoting here from the article:
Maintaining a Reata
The big thing to remember is to keep a reata greased with kidney fat. I have some in the refrigerator that I apply lightly after the reata has been cleaned and dried in the sun. Once the reata has been pulled during initial preparation, it doesn't need to be pulled anymore–using it keeps the reata pulled.
After I get about halfway through a day at a roping, the reata is pretty dirty. If it feels a little dry, I wipe the reata down with a towel and rub a little kidney fat on the reata to soften it. I usually have to work on the first 20 feet because that's where the loop is and it spends more time on the ground.
Every time I use my reata, I clean it with a dry towel to remove the dirt. If it feels dry, I hang it in the sun to warm it and apply a light coat of kidney fat. If it's extremely dirty, I might wipe down my reata quickly with hot, soapy water, allow it to dry completely, and then apply kidney fat. It all depends on how the reata feels. I never over-grease my reata–that makes it heavy, dirty, and lifeless.
And if it's raining, I never use my reata.
Once you have the reata back in shape, it should be fine to use for branding; I've used a reata a lot with smaller calves, and you don't even need to be that proficient at running the reata on your saddle horn if you are sticking to smaller calves that won't pull very hard (I´m certainly not very good at letting the reata slip... which is why I switched to a braided poly rope...). Still, even if you are just planning on roping small calves at a branding, I´d take off the rubber wrapping from your horn; mule hide, rawhide, or smooth leather on your horn will work much better, since these will allow the rope to slip at least a little and prevent even minor jerking. And, they aren't expensive (about $12 from Capriolas and even cheaper at
http://www.animalhealthexpress.com/catalog/product_info.php/products_id/2856). I used saddles with slick metal horns (which slip a lot, and can be an adventure of its own if the horn isn't capped to prevent the dallies from popping up and off the horn...).
Anyway, if the reata has been hanging for an extensive period of time collecting dust, make sure you've cleaned and greased it with kidney fat properly before using it even with smaller calves; a dried out reata will snap pretty easily.
Cheers